Problem #1
:
A differential manometer is used to measure the pressure difference two points in a pipe
of varying cross-sectional area. If the manometric height is
∆
h, determine the velocity at
point 1 (V1) and 2 (V2). Assume frictionless, steady, constant density flow.
Answer:
Assumptions:
a.
Neglect surface tension between the fluid and the mercury (Hg)
b.
Gradual reduction of the pipe between (1) and (2).
Writing the steady flow Bernoulli equation, between (1) and (2)
2
2
1
2
1
1
2
1
1
2
2
f
f
p
p
u
h
u
g
g
ρ
ρ
2
h
+
+
=
+
+
where
1
2
2
1
2
1
(due to assumption (b))
(1)
h
h
A
u
u
A
≈
⎛
⎞
=
⎜
⎟
⎝
⎠
and
are the cross-sectional areas of the pipe at (1) and (2) respectively
A
1
A
2
(
)
1
2
Hg
f
p
p
g
ρ
ρ
h
−
=
−
Δ
Thus
(
)
2
2
1
2
1
2
1
2
f
p
p
u
u
g
ρ
−
−
= −
Or
2
2
2
2
1
1
1
1
2
Hg
f
A
u
h
A
ρ
ρ
⎡
⎤
⎛
⎞
⎛
⎞
−
=
−
⎢
⎥
⎜
⎟
⎜
⎟
⎜
⎟
⎢
⎥
⎝
⎠
⎝
⎠
⎣
⎦
Δ
Or

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